CHAPTER XI.
A TESSARACTIC FIGURE AND ITS PROJECTIONS.
We will now consider a fourth-dimensional shape composed of tessaracts, and the manner in which we can obtain a conception of it. The operation is precisely analogous to that described in chapter VI., by which a plane being could obtain a conception of solid shapes. It is only a little more difficult in that we have to deal with one dimension or direction more, and can only do so symbolically.
We will assume the shape to consist of a certain number of the 81 tessaracts, whose names we have given on p. 168. Let it consist of the thirteen tessaracts: Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo, Vitta, Cura, Penates, Polus, Orcus, Lacerta.
Firstly, we will consider what appearances or projections these tessaracts will present to us according as the tessaractic set touches our space with its (a) Mala cubes, (b) Vesper cubes, (c) Pluvium cubes, or (d) Lar cubes. Secondly, we will treat the converse question, how the shape can be determined when the projections in each of those views are given.
Let us build up in cubes the four different arrangements of the tessaracts according as they enter our space on their Mala, Vesper, Pluvium or Lar sides. They can only be printed by symbolizing two of the directions. In the following tabulations the directions Y, X will at once be understood. The direction Z (expressed by the wavy line) indicates that the floors of nine, each printed nearer the top of the page, lie above those printed nearer the bottom of it. The direction W is indicated by the dotted line, which shows that the floors of nine lying to the left or right are in the W direction (Ana) or the -W direction (Kata) from those which lie to the right or left. For instance, in the arrangement of the tessaracts, as Malas (Table A) the tessaract Tessara, which is exactly in the middle of the eighty-one tessaracts has
| Domitor on its right | side | or in the | X | direction. |
| Ocrea on its left | „ | „ | -X | „ |
| Glans away from us | „ | „ | Y | „ |
| Cudo nearer to us | „ | „ | -Y | „ |
| Sacerdos above it | „ | „ | Z | „ |
| Cura below it | „ | „ | -Z | „ |
| Lacerta in the Ana or | W | „ | ||
| Pallor in the Kata or | -W | „ | ||
Similarly Cervix lies in the Ana or W direction from Urna, with Thyrsus between them. And to take one more instance, a journey from Saltus to Arcus would be made by travelling Y to Remus, thence -X to Sector, thence Z to Mars, and finally W to Arcus. A line from Saltus to Arcus is therefore a diagonal of the set of 81 tessaracts, because the full length of its side has been traversed in each of the four directions to reach one from the other, i.e. Saltus to Remus, Remus to Sector, Sector to Mars, Mars to Arcus.
TABLE A.
Mala presentation of 81 Tessaracts.
| Z | W | -W | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Arcus | Ovis | Portio | Ara | Vomer | Pluma | Mars | Merces | Tyro | |||||||||||||||||
| Laurus | Tigris | Segmen | Praeda | Sacerdos | Hydra | Spicula | Mora | Oliva | |||||||||||||||||
| Axis | Troja | Aries | Cortex | Mica | Flagellum | Comes | Tibicen | Vestis | |||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Postis | Clipeus | Tabula | Pilum | Glans | Coins | Ala | Cortis | Aer | |||||||||||||||||
| Orcus | Lacerta | Testudo | Ocrea | Tessera | Domitor | Uncus‡ | Pallor‡ | Tergum | |||||||||||||||||
| Verbum | Luctus | Anguis | Cardo | Cudo | Malleus | Ostrum | Bidens‡ | Scena | |||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Telum | Nepos | Angusta | Agmen | Lacus | Arvus | Sector | Hama | Remus | |||||||||||||||||
| Polus | Penates | Vulcan | Crates | Cura | Limen | Frenum‡ | Plebs‡ | Sypho | |||||||||||||||||
| Cervix | Securis | Vinculum | Thyrsus | Vitta | Sceptrum | Urna‡ | Moles‡ | Saltus | |||||||||||||||||
| -Z | X | X | X | ||||||||||||||||||||||
TABLE B.
Vesper presentation of 81 Tessaracts.